Transcript Graphing Trig Functions

Unit 7: Trigonometric Functions
Graphing the Trigonometric Function
E.Q: E.Q
1. What is a radian and how do I use it to
determine angle measure on a circle?
2. How do I use trigonometric functions to
model periodic behavior?
CCSS: F.IF. 2, 4, 5 &7E; F.TF. 1,2,5 &8
Mathematical Practices:
 1. Make sense of problems and persevere in solving them.
 2. Reason abstractly and quantitatively.
 3. Construct viable arguments and critique the reasoning of
others.
 4. Model with mathematics.
 5. Use appropriate tools strategically.
 6. Attend to precision.
 7. Look for and make use of structure.
 8. Look for and express regularity in repeated reasoning.
SOH
CAH
TOA
CHO
SHA
CAO
Right Triangle Trigonometry
Graphing the Trig Function
Graphing Trigonometric Functions

Amplitude: the maximum or minimum vertical
distance between the graph and the x-axis.
Amplitude is always positive
5
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
4
y = 2 sin x
y = 4 sin x
Graphing Trigonometric
Functions

Period: the number of degrees or radians we must
graph before it begins again.
7
The period of a function is the x interval needed for the
function to complete one cycle.
For b  0, the period of y = a sin bx is 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y  sin 2
period: 2
period: 
y  sin x x


2
If b > 1, the graph of the function is shrunk horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
The sine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where y = sin θ. As the particle moves
through the four quadrants, we get four pieces of the sin graph:
I. From 0° to 90° the y-coordinate increases from 0 to 1
II. From 90° to 180° the y-coordinate decreases from 1 to 0
III. From 180° to 270° the y-coordinate decreases from 0 to −1
IV. From 270° to 360° the y-coordinate increases from −1 to 0
sin θ
y
90°
135°
45°
II
I
I
0°
180°
II I
II
x
0
90°
180°
IV
225°
III
315°
270°
Interactive Sine Unwrap
360° θ
270°
IV
θ
sin θ
0
0
π/2
1
π
0
3π/2
−1
2π
0
Sine is a periodic function: p = 2π
sin θ
−3π
−2π
−π
0
π
2π
One period
2π
sin θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
sin θ is an odd function; it is symmetric wrt the origin.
  Domain, sin(−θ) = −sin(θ)
3π θ
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2
1

3
2
2
5
2
x
The cosine function
Imagine a particle on the unit circle, starting at (1,0) and rotating
counterclockwise around the origin. Every position of the particle
corresponds with an angle, θ, where x = cos θ. As the particle moves
through the four quadrants, we get four pieces of the cos graph:
I. From 0° to 90° the x-coordinate decreases from 1 to 0
II. From 90° to 180° the x-coordinate decreases from 0 to −1
III. From 180° to 270° the x-coordinate increases from −1 to 0
IV. From 270° to 360° the x-coordinate increases from 0 to 1
y
cos θ
90°
135°
45°
II
I
I
0°
180°
II I
IV
225°
315°
270°
IV
x
0
90°
270°
180°
II
III
θ
360°
θ
cos θ
0
1
π/2
0
π
−1
3π/2
0
2π
1
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
cos x
1
2
0
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = cos x
y
3

2



1

2
2
1

3
2
2
5
2
x
Cosine is a periodic function: p = 2π
cos θ
θ
−3π
−2π
−π
0
π
One period
2π
cos θ: Domain (angle measures): all real numbers, (−∞, ∞)
Range (ratio of sides): −1 to 1, inclusive [−1, 1]
cos θ is an even function; it is symmetric wrt the y-axis.
  Domain, cos(−θ) = cos(θ)
2π
3π
Properties of Sine and Cosine graphs
1. The domain is the set of real numbers
2. The rage is set of “y” values such that -1≤ y ≤1
3. The maximum value is 1 and the minimum value
is -1
4. The graph is a smooth curve
5. Each function cycles through all the values of the
range over an x interval or 2π
6. The cycle repeats itself identically in both
direction of the x-axis
15
Given : A sin Bx
Sine Graph
 Amplitude = IAI
 period = 2π/B
Example:
y=5sin2X
› Amp=5
π/2
π/4
› Period=2π/2
=π
3π/4
π
Given : A sin Bx
 Cosine Graph
 Amplitude = IAI
 period = 2π/B
 Example:
y=2cos 1/2 X
› Amp= 2
2π
› Period= 2π/(1/2)
4π
π
3π
4π
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
x
y = 3 cos x
y

(0, 3)
2
1

0
3
2
0
max

-3
x-int min
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
2
( 3 , 0)
2
3
4 x
Use basic trigonometric identities to graph y = f (–x)
Example : Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
Example : Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
b
Calculate the five key points.
x
0
y = –2 sin 3x
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2

(  ,-2)
6
2
2
3

2
(  , 0) 2
3
( , 0)
3
5
6

x
Tangent Function
Recall that tan  
sin 
.
cos 
Since cos θ is in the denominator, when cos θ = 0, tan θ is undefined.
This occurs @ π intervals, offset by π/2: { … −π/2, π/2, 3π/2, 5π/2, … }
Let’s create an x/y table from θ = −π/2 to θ = π/2 (one π interval),
with 5 input angle values.
θ
sin θ
cos θ
tan θ
θ
tan θ
−π/2
−1
0
und
−π/2
und
2
2
2
2
−1
−π/4
−1
0
0
1
0
0
0
π/4
2
2
2
2
1
π/4
1
π/2
1
0
und
π/2
und
−π/4

Graph of Tangent Function:
Periodic
Vertical asymptotes
tan θ
where cos θ = 0
tan  
θ
−π/2
tan θ
Und (-∞)
−π/4
−1
0
0
π/4
1
π/2
Und(∞)
−3π/2
−π/2
0
π/2
One period: π
tan θ: Domain (angle measures): θ ≠ π/2 + πn
Range (ratio of sides): all real numbers (−∞, ∞)
tan θ is an odd function; it is symmetric wrt the origin.
  Domain, tan(−θ) = −tan(θ)
3π/2
θ
sin 
cos 
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. Domain : all real x

x  k  k   
2
2. Range: (–, +)
3. Period: 
4. Vertical asymptotes:

x  k  k   
2

2
 3
2

2
period: 
3
2
x
Example: Find the period and asymptotes and sketch the graph
 y

1
x


x

of y  tan 2 x
4
4
3
1. Period of y = tan x is  .

 Period of y  tan 2 x is .
3
 1
2


, 

8
2
 8 3
x
2. Find consecutive vertical
 1
asymptotes by solving for x:
 3 1 
 , 
 , 


 8 3
 8 3
2x   , 2x 
2
2


Vertical asymptotes: x   , x 
4
4


 3
3. Plot several points in (0, )

x
0
2
8
8
8
1
1
1
1

y  tan 2 x 
0
4. Sketch one branch and repeat.
3
3
3
3
Cotangent Function
cos 
.
sin
Since sin θ is in the denominator, when sin θ = 0, cot θ is undefined.
Recall that cot  
This occurs @ π intervals, starting at 0: { … −π, 0, π, 2π, … }
Let’s create an x/y table from θ = 0 to θ = π (one π interval),
with 5 input angle values.
θ
sin θ
cos θ
cot θ
θ
cot θ
0
Und ∞
0
1
Und ∞
π/4
2
2
2
2
1
π/4
1
π/2
1
0
0
π/2
0
3π/4
2
2
−1
3π/4
−1
π
Und−∞
0
π
0

2
2
–1
Und−∞
Graph of Cotangent Function: Periodic
Vertical asymptotes
where sin θ = 0
cos 
cot  
sin
cot θ
θ
cot θ
0
∞
π/4
1
π/2
0
3π/4
−1
π
−∞
−3π/2
-π
−π/2
π/2
cot θ: Domain (angle measures): θ ≠ πn
Range (ratio of sides): all real numbers (−∞, ∞)
cot θ is an odd function; it is symmetric wrt the origin.
  Domain, tan(−θ) = −tan(θ)
π
3π/2
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot x 
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y  cot x
1. Domain : all real x
x  k k  
2. Range: (–, +)
3. Period: 
4. Vertical asymptotes:
x  k k  
vertical asymptotes

3
2
 

2
x  
x0

 3
2
2
x 
x
2
x  2
Cosecant is the reciprocal of sine
Vertical asymptotes
where sin θ = 0
csc θ
−3π
θ
0
−2π
−π
π
2π
3π
sin θ
One period: 2π
sin θ: Domain: (−∞, ∞)
Range: [−1, 1]
csc θ: Domain: θ ≠ πn
(where sin θ = 0)
Range: |csc θ| ≥ 1
or (−∞, −1] U [1, ∞]
sin θ and csc θ
are odd
(symm wrt origin)
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x 
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y  csc x
1. domain : all real x
x  k k  
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
x  k k  
where sine is zero.
x



2
2

3 2
2
5
2
y  sin x
4
Secant is the reciprocal of cosine Vertical asymptotes
where cos θ = 0
sec θ
θ
−3π
−2π
−π
0
π
2π
3π
cos θ
One period: 2π
cos θ: Domain: (−∞, ∞) sec θ: Domain: θ ≠ π/2 + πn
(where cos θ = 0)
Range: [−1, 1]
Range: |sec θ | ≥ 1
or (−∞, −1] U [1, ∞]
cos θ and sec θ
are even
(symm wrt y-axis)
Graph of the Secant Function
1
sec
x

The graph y = sec x, use the identity
.
cos x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y  sec x
Properties of y = sec x
1. domain : all real x

x  k  (k  )
2
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2
4
y  cos x
x



2
2
4

3
2
2
5
2
3
Summary of Graph Characteristics
Def’n
∆
sin θ
csc θ
cos θ
sec θ
tan θ
cot θ
о
Period
Domain
Range
Even/Odd
Summary of Graph Characteristics
Def’n
Period
Domain
Range
Even/Odd
−1 ≤ x ≤ 1 or
[−1, 1]
odd
∆
о
sin θ
opp
hyp
y
r
2π
(−∞, ∞)
csc θ
1
.sinθ
r
.y
2π
θ ≠ πn
cos θ
adj
hyp
x
r
2π
(−∞, ∞)
sec θ
1 .
sinθ
r
y
2π
θ ≠ π2 +πn
tan θ
sinθ
cosθ
y
x
π
θ ≠ π2 +πn
All Reals or
(−∞, ∞)
odd
cot θ
cosθ
.sinθ
x
y
π
θ ≠ πn
All Reals or
(−∞, ∞)
odd
|csc θ| ≥ 1 or
(−∞, −1] U [1, ∞)
All Reals or
(−∞, ∞)
|sec θ| ≥ 1 or
(−∞, −1] U [1, ∞)
odd
even
even
14. 2: Translations of Trigonometric Graphs
•Without looking at your notes, try to sketch the basic shape of
each trig function:
1) Sine:
2) Cosine:
3) Tangent:
More Transformations
We have seen two types of transformations on trig
graphs: vertical stretches and horizontal stretches.
There are three more: vertical translations (slides),
horizontal translations, and reflections (flips).
More Transformations
Here is the full general form for the sine function:
y  k  a sin bx  h
Just as with parabolas and other functions, h and k
are translations:

h slides the graph horizontally (opposite of sign)

k slides the graph vertically
Also, if a is negative, the graph is flipped vertically.
More Transformations
 To graph a sine or cosine graph:
1.
Graph the original graph with the correct
amplitude and period (like section 14.1).
2. Translate h units horizontally and k units
vertically.
3. Reflect vertically at its new position if a is negative
(or reflect first, then translate).
Examples
 Describe how each graph would be transformed:
1.
y  2  sin x
2.


y  cos x  
2

3.
y  2  sin( x   )
Examples
 State the amplitude and period, then graph:
y  2  cos( x)
x
-2π
2π
Examples
 State the amplitude and period, then graph:


y   sin  x  
2

x
-2π
2π

Examples
State the amplitude and period, then graph:
1
y  2  sin x
2
x
-2π
2π
Examples
 Write an equation of the graph described:

The graph of y = cos x translated up 3 units,
right π units, and reflected vertically.
14.3: trigonometric Identities
 Reciprocal Identities
 Quotient Identities
 Pythagorean Identities
 Opposite Angles Identity
Some Vocab
Identity: a statement of equality between two
expressions that is true for all values of the
variable(s)
2. Trigonometric Identity: an identity involving trig
expressions
3. Counterexample: an example that shows an
equation is false.
1.
Prove that sin(x)tan(x) = cos(x) is not a trig
identity by producing a counterexample.
 You can do this by picking almost any angle
measure.
 Use ones that you know exact values for:
 0, π/6, π/4, π/3, π/2, and π
Reciprocal Identities
Quotient Identities
Why?
Do you remember the Unit Circle?
 What is the equation for the unit circle?
x2 + y2 = 1
• What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean
Identity!
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1 .
cos2θ cos2θ cos2θ
tan2θ + 1 = sec2θ
Quotien
t
Identity
another
Pythagorean
Identity
Reciproc
al
Identity
Take the Pythagorean Identity and
discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1 .
sin2θ sin2θ sin2θ
1 + cot2θ = csc2θ
Quotien
t
Identity
a third
Pythagorean
Identity
Reciproc
al
Identity
Opposite Angle Identities
sometimes these are called even/odd identities
Simplify each expression.
Using the identities you now know, find
the trig value.
If cosθ = 3/4,
find secθ.
If cosθ = 3/5,
find cscθ.
sinθ = -1/3, 180o < θ < 270o; find tanθ
secθ = -7/5, π < θ < 3π/2; find sinθ
– Similarities and Differences
a) How do you find the
amplitude and period for
sine and cosine functions?
b) How do you find the
amplitude, period and
asymptotes for tangent?
c) What process do you
follow to graph any of the
trigonometric functions?